Quantum mechanical ab initio calculations of corner- and edge-sharing
cluster interlinking in binary As/Ge-S glass formers were performed using
HyperChem Realise 7,5 program. The average formation energies
(AFE) of clusters with respect to single pyramid-like AsS3/2 and
tetragon-like GeS4/2 building blocks were calculated. It was shown
that AFE associated with the number of Lagrangian constraints per atom would
explain the adaptability of chalcogenide glass-forming backbone to structural
self organization.

Key words: Chalcogenide glasses, ab initio calculations.

The chalcogenide
glasses (ChG) derive their name from chalcogen (Ch) elements S, Se or Te but
not O as a main component in their chemical composition. They are important
materials both for fundamental investigation of particularities in the
disordered state and for different applications. Two common chalcogenide
systems consist of V-VI elements, i.e. compounds with pnictogen (P) atoms from
V group (e.g. As, Sb), where the glass-forming regions are mostly centered
around stoichiometric P2Ch3 composition,
and compounds with group IV elements, i.e. Si, Ge (denoted tetragens, T, in
view of the prevalent tetrahedral coordination), where binary glass-forming
IV-VI compositions are centered around stoichiometric TCh2 composition. The atoms in ChG form extended three dimensional
networks, they maintaining short-range order by keeping the number of covalent
bonds to nearest neighbours in strong dependence on the valence of constituent
atoms. Hence group IV elements are four-fold, group V elements are three­fold
and group VI elements are two-fold coordinated.

Amorphous
chalcogenides of arsenic and germanium are the most characteristic materials.
It is well know that at stoichiometricAs2S3 and GeS2
compositions (i.e. at the compositions where only heteropolar chemical bonds
exist) one As atom is linked to three S atoms and each S atom is linked to two
As atoms in As-S glass or each Ge atom is linked to four S atoms and each S
atom is linked to two Ge atoms in Ge-S. Experimental data suggest the presence
of following structural units in a glass-forming network - pyramidal
As(S1/2)3 with As atom raised above the plane defined by
three S atoms and tetrahedral Ge(Se1/2)4 being a
tetrahedron centred on Ge atoms [1].

In dependence on
chemical composition the ChG change the number of Lagrangian

constraints per atom nc forming under- (nc<3),
over- (nc>3) or optimally-constrained (nc=3)
atomic networks with fully saturated covalent bonding. It was assumed by
Phillips [2] that optimal mechanical stability of the
network can be achieved when nc=3, this network being called self-organized
phase. The underconstrained (floppy) network with nc<3
is easily deformed, but in the overconstrained (rigid) networks with nc>3
any deformation requires stretching or bending bonds. The bonds are not
distributed randomly and the network can adapt itself to lower the stress due
to overconstrained regions. In general, the nc number can
be calculated according to the known mean-field constraints theory [3, 4]:

= -+ (2Z-2V

■3) + —-

N

(1)

where: Z is coordination number of glass network constructed by nr atoms

In this work, we have
used new cation-interlinking network cluster approach (CINCA) to built the
simplest molecular-like species with fully-saturated covalent bonding within
binary As/Ge-S systems: pyramid-like AsS3/2 building blocks having 3 shared S
atoms and tetragon-like GeS4/2 building blocks having 4 shared S
atoms. These basic structural units having 2 cations (As or Ge) create the
whole ChG network. They can be interconnected into two ways, especially as
atom-shared clusters (ASC) with shared S atom or as bond-shared clusters (BSC)
with shared S-S bond. Figure 1 shows ASC (a) and BSC (b) for two pyramidal units. The ending
chalcogen atoms belong to basic unit in the case of BSC cluster, while we
should consider only half-part contribution of each terminal chalcogen atom in
the case of ASC (because the next half takes part in another unit).

ASC

BSC

a b Fig.
1. ASC cluster interlinked with
S atom (a) and BSC cluster interlinked with S-S bond (b) for two pyramidal As(S1/2)3
units (the black color is for S atoms and grey one for As atoms)

The calculations only
for ASC structural units were performed. We have studied the formation energy
of different type of interconnections between building structural units:
corner-sharing CS, edge-sharing ES or face-sharing FC, either pyramidal or

M. Hyla, V.
Boyko, О. Shpotyuk et al.

tetrahedral basic
units. As a point of reference, energies of single As(S1/2)3
building blocks having 3 shared chalcogen atoms (Z= 2,40) and Ge(S1/2)4
building blocks having 4 shared chalcogen atoms (Z = 2,67). All
clusters are shown in fig. 2 and
3.

c d

Fig. 2. Schematic
view of geometrically optimized single (a), corner- (b), edge- (c) and
face-shared (d) As(S1/2)3 units (the black color is for S
atoms and grey one for As atoms)

Recently, the
first-principles methods have been very successful to calculate structural
properties of materials. In order to study the optimal geometries and calculate
formation energy of clusters ab initio method
based on the Hartree-Fock approximation was used. A few ab initio calculations of clusters energy were carried
out by R.M. Holomb and co-workers [5-7]. Although the calculations do not allow to
drive a conclusion about preferences in connection suitable building units.
Then, a new energetic parameter, an average formation energy (AFE), which is
defined as formation energy per atom in a cluster, was introduced in our
calculations. We have made our conclusions on the base of this parameter. In
addition, our analysis was supplemented by mean-field constraints theory.

Quantum mechanical
calculations were performed using HyperChem Realise 7,5 PC
program [8]. The RHF ab initio level was taken as a ground for mathematical
calculating procedure, either geometry optimization or single point energy, the
6-311G* basis set, being employed. All ending S atoms within clusters were
additionally terminated by H atoms to be two-fold coordinated in full respect
to their saturated covalent bonding. As a result of single point energy
calculations, the total energy Et for each molecule was obtained. Since all clusters were of ASC type,
the V energy of S atom along with energy of H atoms and bonds between them were
subtracted from total energy Et. The ASC energy was calculated as half of total energy of H-S-H molecule giving EASC= -125091,0876 kcal/mol. The energy of single As, Ge and S
atoms was accepted to be EAs= -1401900,133 kcal/mol, EGe=
-1302219,833 kcal/mol and ES=
-249381,9706 kcal/mol,
respectively. The overall energy of atoms within cluster Eat was equal to:

where nAs, nGe and nS are the number of As, Ge and S atoms, respectively. The cluster
formation energy, Ef, was calculated as:

Ef = Ec - Eal, (3) where Ec was total
energy of the cluster, Eat was overall energy of atoms forming the cluster.

To estimate the
glass-forming tendency within cation-interlinked clusters, the average
formation energy AFE defined as cluster formation energy per atom was
introduced:

Ea/=Ef/ N, (4) where Ef was cluster
formation energy in respect to (3), Nwas total number of atoms in the cluster (N = nAs + nGe + nS).

The structures of
clusters studied displayed in fig. 2 and 3 (the terminated H atoms are not
shown).

The optimized bond distances
and bond angles for all clusters are given in table 1 and 2. These values are
quite close to known experimental data proper to crystalline As2S3
and GeS2. We calculated the number of Lagrangian constraints per
atom, nc, too. The results of both constraints and
energetic calculations for both binary systems (As-S and Ge-S) are presented in
Table 3 and 4, respectively.

Table 1

Geometric parameters
of optimized As(S1/2)3-based clusters

Bond distance

Bond angle

Type

[10-4 nm]

[deg]

As-S

S-As-S

As-S-As

2251,5

101,94

As(Si/2)3-single

2254,0

102,94

-

2256,1

92,67

2248,6

102,62

98,9

2252,3

103,17

As(S1/2)3-CS

2263,5 2264,3

92,35 96,98

2252,4

99,57

2253,1

97,14

2252,2

101,00

91,18

2266,7

101,60

91,16

As(Si/2)3-ES

2266,3 2267,4

88,83 101,46

2267,0

100,89

2250,6

88,80

2284,8

86,99

74,74

2284,9

86,98

74,74

As(Si/2)3-FS

2284,6

86,99

74,74

2284,8

86,99

2284,8

86,98

2284,7

86,99

Table 2

Geometric parameters
of optimized Ge(S1/2)4-based clusters

Bond distance

Bond angle

Type

[10-4
nm]

[deg]

Ge-S

S-Ge-S

Ge-S-Ge

2224,9

112,67

2236,6

109,62

Ge(S1/2)4-single

2236,6 2215,1

109,42 102,80

-

112,67

109,42

2238,5

108,56

98,42

2238,7

107,88

2238,5

110,67

2242,7

108,28

2238,5

109,53

Ge(S1/2)4-CS

2238,6 2238,5

111,82 110,67

2242,8

111,83

108,55

107,88

109,53

108,28

2224,0

111,88

83,79

2223,9

113,88

84,30

2231,5

110,06

2242,4

110,05

2224,0

113,88

Ge(S1/2)4-ES

2223,9 2242,4

95,96 110,05

2231,4

113,88

111,88

110,05

113,88

95,96

2194,9

124,80

67,62

2274,9

124,80

67,62

2274,9

121,89

67,81

2269,7

91,76

2194,9

92,13

Ge(S1/2)4-FS

2275,6 2275,6

92,13 124,91

2269,7

124,91

121,71

92,11

92,11

91,72

Table 3

Results of calculated
the number of Lagrangian constraints per atom, nc,.and energetic
calculations

for both binary
system As-S; Z=2,40

Cluster
characterization

Energetic
parameters

Type

N

nc

AsnSmHp

total energy, Et

AsnSm

cluster

energy,Ec

Forma­tion energy,

Ef =Ec -

AFE

average formation

energy,

Efav-Ef/N

kcal/mol

kcal/mol

kcal/mol

kcal/mol

As(S1/2)3 -single

2,5

3,00

-2151444,860

-1776171,598

-198,509

-79,404

As(S1/2)3 -CS

5

3,00

-4052707,566

-3552343,215

-397,038

-79,408

As(S1/2)3 -ES

5

2,60

-3802519,623

-3552337,448

-391,271

-78,254

As(S1/2)3 -FS

5

2,20

-3552319,113

-3552319,113

-372,936

-74,587

Table 4

Results of calculated
the number of Lagrangian constraints per atom, nc,.and energetic
calculations

for both binary
system Ge-S; Z=2,67

Cluster
characterization

Energetic
parameters

Type

N

nc

GenSmHp

total energy,

Et

GenSm

cluster

energy,Ec

Forma­tion energy,

Ef = Ec
-

Eat

AFE average
formation

energy,

Efav-Ef/N

kcal/mol

kcal/mol

kcal/mol

kcal/mol

Ge(S1/2)4

-single

3

3,67

-2301637,257

-1801272,907

-289,133

-96,378

Ge(S1/2)4

-CS

6

3,67

-4353080,104

-3602533,578

-566,029

-94,338

Ge(S1/2)4

-ES

6

3,00

-4102907,744

-3602543,394

-575,845

-95,974

Ge(S1/2)4

-FS

6

3,00

-3852684,078

-3602501,903

-534,354

-89,059

In the case of
pyramidal As(S1/2)3-based clusters, the highest AFE value
(-74,587 kcal/mol) is achieved for face-sharing interlinking (table 3). It means
that these structural units are unprofitable in ChG from energetic point of
view. The AFE either of single As(S1/2)3 pyramid
or two corner-sharing As(S1/2)3 pyramids
are the lowest ones being as high as
-79,404 kcal/mol and -79,408
kcal/mol, respectively. Hence the structure of binary As-S system prefers units
built of corner-sharing As(S1/2)3 pyramids.

These corner-sharing
pyramids form the optimally-constrained atomic network (nc=3), while the edge-sharing pyramids form only under-constrained
glassy network (nc<3).

Therefore, on the
basis of our calculations, we can expect the corner-sharing pyramidal-like
interlinking in stoichiometric As2S3 ChG in full respect to known structural model given by Zachariasen
yet in the 1930-s [9].

In the case
tetrahedral Ge(S1/2)4-based clusters, the highest AFE value
(-89,059 kcal/mol) is achieved for face-sharing interlinking too (see table 4).
So these structural units apparently do not occur in the real glass-forming
networks. The lowest AFE value (-96,378 kcal/mol) is character for single Ge(S1/2)4 cluster and two-cation interlinking edge-sharing cluster (-95,974
kcal/mol), while the corner-sharing tetrahons

have AFE=-94,338
kcal/mol.

Therefore, the
edge-sharing tetrahedral-like units form the optimally-constrained glassy
network (nc=3), while
the corner-sharing units form the over-constrained one (nc>3). In
general, this conclusion appears to be consistent with outrigger raft model
proposed by Phillips [10].

On the basis of
calculations performed with HyperChem Realise 7,5 program, it is shown that
corner-shared As(S1/2)3 pyramids form the optimally-constrained
atomic network (nc=3) with the
lowest AFE value, while the edge-sharing interlinking of Ge(S1/2)4 tetragons is the most preferential one for optimally-constrained
Ge-S glassy network.